The boiling curve that turns back

The heat that changes no temperature found where the latent heat of boiling goes — into pulling molecules apart, not into speeding them up. A boiling point is a pressure found that the temperature at which water boils is set by the pressure above it. The barrier a new phase has to climb found why a bubble does not form the moment the boiling point is reached: a small bubble costs surface energy, and the liquid must be superheated, or offered a crevice, before one can grow. The part of the curve no fluid follows found an unstable branch in the equation of state itself, and the melting line that turns round followed water’s phase boundary to extreme pressures.

All of those concern equilibrium: what boils, where and at what temperature. This essay concerns the rate. When a hot surface meets water, how much heat does it pass, and how does that depend on how hot the surface is? The obvious answer — more heat for a hotter surface — is right at first and then spectacularly wrong. The dependence rises, peaks, falls by a factor of fifty and rises again, and the falling part governs the safety of every system that uses boiling water to carry heat away.

A hotter wire that boiled less #

In 1934 Shigeo Nukiyama, a Japanese engineer, passed an electric current through a thin wire immersed in water at its boiling point and measured the heat it gave off against its temperature, read from its electrical resistance. As he raised the current the wire got hotter and boiled more vigorously, and the heat it passed rose steeply. Then, at a certain power, the wire suddenly glowed red and, with a platinum wire, survived at a far higher temperature; with nichrome it melted. Nukiyama realised that the heat flux as a function of wire temperature could not be simply increasing: there had to be a range in which a hotter wire passed less heat, which his power-controlled experiment could not follow and which it jumped over. Three years later Thomas Drew and Alfred Mueller traced that range directly by fixing the surface’s temperature instead of its power, using condensing vapours of different boiling points to heat a tube. The result is the boiling curve.

The curve #

The figure assembles the curve from the standard correlations for water at one atmosphere on a horizontal surface. At a few degrees above boiling no bubbles form; heat leaves by ordinary convection, the warmed water rising and being replaced, and the flux grows gently with the superheat. From about five degrees, bubbles nucleate at pits and scratches in the surface, grow, detach and rise. This is nucleate boiling, the regime of a kettle, and the flux climbs steeply — as the cube of the superheat in the classic correlation — because every extra degree activates more nucleation sites and makes each bubble grow faster.

At about twenty degrees above boiling the flux reaches its maximum, 1.1 megawatts per square metre: the critical heat flux. Beyond it the curve turns over. The bubbles are now so numerous and so close that they merge into columns and patches of vapour, and the vapour, which conducts heat far worse than liquid, begins to separate the surface from the water. The hotter the surface, the more of it is covered, and the flux falls — the transition regime — until, at a surface temperature the correlations here place about 67 degrees above boiling, the surface is wholly covered by a continuous film of vapour. That is the Leidenfrost point, where the flux has fallen to about 19 kilowatts per square metre. Beyond it, in film boiling, heat must cross the vapour film by conduction and radiation, and the flux rises again slowly as the surface gets hotter, reaching the critical value again only at well over a thousand degrees.

Between the crisis and the Leidenfrost point, a surface three times hotter passes fifty-eight times less heat.

Why the bubbles start where they do #

The rising branch owes its steepness to the surface’s imperfections. The barrier a new phase has to climb found that a bubble cannot simply appear in liquid at its boiling point: a bubble of radius $r$ holds its vapour at a pressure raised by $2\sigma/r$, so a very small bubble needs liquid well above the boiling point to survive, and pure water in a perfectly smooth, clean vessel can be superheated by tens of degrees before it boils explosively. Real surfaces are not smooth. Pits and scratches a few micrometres across trap tiny pockets of gas or vapour that never fully fill with liquid — the geometry that also lets a pore fill from dry air, with the curvature working the other way — and each pocket is a ready-made bubble of the pocket’s radius, needing only a superheat of a few degrees to grow.

As the surface heats, smaller and smaller pockets become active, because the superheat needed falls as the pocket’s radius grows and rises as it shrinks. A surface has many more small pits than large ones, so the number of active sites climbs rapidly with superheat, and with it the flux. That is the origin of the steep cube law in the nucleate regime, and why the constant in front of it depends on the surface: a sandblasted surface boils more readily than a polished one, a clean surface differently from one coated with scale, and a surface that liquid wets well floods its pockets and deactivates them. The same physics sets a practical rule in laboratories and kitchens: a boiling chip or a scratched pot prevents the sudden, violent boiling of superheated water, by providing sites where bubbles can start gently.

Why the crisis comes #

The crisis is a problem of traffic. Heat leaves the surface as vapour, and vapour must leave as fast as it is made, rising away while liquid comes in to replace it. The two streams pass each other near the surface: columns of vapour going up, liquid coming down between them. Novak Zuber argued in 1959 that the arrangement fails when the vapour columns become unstable — when the counterflowing streams of vapour and liquid can no longer pass without the interface between them breaking up, a Helmholtz instability, on a spacing set by the balance of surface tension against buoyancy, a Rayleigh–Taylor instability. The maximum heat flux the traffic can carry is then

$$q_{\max} = 0.131\, h_{fg}\,\rho_v^{1/2}\,\big[\sigma g (\rho_l - \rho_v)\big]^{1/4},$$

which for water at one atmosphere is 1.1 megawatts per square metre. The formula contains the latent heat, because that sets how much heat each kilogram of vapour carries; the vapour density, because a dense vapour carries more heat per unit volume; the surface tension and gravity, which set the spacing of the vapour columns; and nothing about the surface itself. That last point is only approximately true — surfaces that wet well and have many nucleation sites reach higher crisis fluxes — but the formula’s success at predicting the crisis from the fluid’s properties alone is why it is still used.

Why the minimum comes #

The Leidenfrost point is the mirror image. Coming from high temperatures, the surface is covered by a stable film of vapour, and liquid never touches it. As the surface cools the film thins and the vapour it generates slows, until at some temperature the film can no longer be sustained: liquid touches the surface somewhere, boils explosively, and the film collapses. Zuber’s analysis of the minimum again rests on the stability of the interface between the vapour film and the liquid above it, which is a heavier fluid resting on a lighter one — unstable to Rayleigh–Taylor waves — held up only by the vapour’s continual generation. It gives a minimum flux of about 19 kilowatts per square metre for water, independent of the surface.

The temperature at which the minimum occurs is much less universal. It depends strongly on the surface’s roughness, its thermal properties and how easily liquid wets it, and measured Leidenfrost temperatures for water range from about 150 to over 300 °C; on polished metal they are near 200. The correlations here put it at 167, a reminder that the transition regime between the two stable branches is the least reliably known part of any boiling curve, drawn dashed in the figure for that reason.

Setting the power, not the temperature #

The falling branch has a practical consequence that is also Nukiyama’s observation.

Most practical heaters fix the power, not the temperature: an electric element, a fuel rod in a reactor, a furnace wall. The surface’s temperature is then whatever the boiling curve gives at that heat flux. As long as the flux is below the crisis, there is a stable operating point on the nucleate branch, twenty degrees or less above boiling. Push the flux past the crisis and that point disappears. The only other point on the curve at that flux lies on the film branch, far to the right, and the surface jumps there — in these correlations, to over 1,700 degrees above boiling, past the melting point of copper and of steel. That is burnout, or boiling crisis, or departure from nucleate boiling in the language of reactor engineering, and it can happen with the tube full of water.

The jump is not reversed by reducing the power back below the crisis. The surface stays on the film branch, insulated by its vapour, until the flux falls below the Leidenfrost minimum, and only then drops back to nucleate boiling. The boiling curve under set power therefore has hysteresis: the path up and the path down differ, and between them lies the transition branch, which is never visited. That branch has a negative slope — more heat for a cooler surface — and a state on it is unstable under set power in the same way that the part of the curve no fluid follows, the stretch of the van der Waals isotherm on which pressure rises as volume increases, is unstable in a fluid: any small disturbance grows. Only by fixing the temperature, as Drew and Mueller did, can the branch be held and measured.

The drop that lasts longest on the hottest pan #

The same curve explains a kitchen observation that looks like a paradox.

Flick water onto a pan just above boiling and the drops simmer slowly; onto a hotter pan and they hiss and vanish in a second or two; onto a very hot pan and they bead up and skate about for a minute or more. The lifetime of a drop is roughly its latent heat divided by the heat flowing into it, so it follows the boiling curve upside down: long where the flux is small, shortest at the crisis, and long again at the Leidenfrost point, where the drop floats on a cushion of vapour a fraction of a millimetre thick, generated by its own evaporation and insulating it from the metal. The figure makes the estimate with the correlations above, and the shape — a short lifetime at the crisis between two long ones — is what careful measurements find, though the actual numbers depend on the pan’s surface and the drop’s size.

Johann Gottlob Leidenfrost described the effect in 1756, watching drops on a red-hot spoon. It is the reason a cook tests a pan by flicking water onto it — a pan hot enough to make drops dance is hot enough to sear — and it is the reason molten metal workers, and demonstrators, can briefly dip a wet hand into molten lead without harm, protected for a moment by a film of steam, a demonstration nobody should repeat.

The best way known to move heat, and its collapse #

Dividing the flux by the superheat gives the heat-transfer coefficient, the conductance of the boundary between surface and water. Nucleate boiling is the most effective way known to move heat from a surface into a fluid: at the crisis the coefficient is fifty-five thousand watts per square metre per kelvin, because each departing bubble carries off latent heat, far more per kilogram than warming the liquid could, and each departure pulls fresh, cooler liquid onto the surface behind it. At the Leidenfrost point the coefficient has collapsed to under three hundred, nearly two hundred times worse, because heat must now cross a layer of vapour whose thermal conductivity is about that of air.

That combination — the best heat transfer there is, bordered by a cliff — is why boiling is used wherever large amounts of heat must be removed from a small area, and why it is always used with a margin. A pressurised-water reactor’s fuel rods, the tubes of a power station’s boiler and the cooling channels of high-power electronics are all designed around a ratio between the local heat flux and the critical heat flux predicted for that spot, kept well above one everywhere, because the consequence of crossing it is not a gradual loss of performance but a jump.

The crisis that gravity postpones #

The crisis depends on gravity, through the buoyancy that lifts vapour away. Zuber’s formula makes the critical flux proportional to the fourth root of gravity: on the Moon it would be two-thirds of its terrestrial value, on Mars four-fifths. In an orbiting spacecraft, at a ten-thousandth of Earth’s gravity, it would be a tenth — and the formula, derived for buoyancy-driven flow, is itself unreliable there. Experiments in drop towers, parabolic flights and on the International Space Station find that in near-weightlessness a bubble does not detach but stays where it forms, grows and merges with its neighbours until a single large bubble covers the heater, and the crisis comes at a small fraction of the terrestrial flux. Spacecraft that use boiling to reject heat must pump the liquid past the surface to do the job buoyancy does on the ground.

Where the correlations stop #

Every curve in these figures comes from empirical correlations, and each has a limited range. The nucleate-boiling correlation contains a constant that depends on the particular combination of liquid and surface and can vary by a factor of two or three between surfaces; the transition regime is an interpolation, because its real shape depends on how liquid and vapour contact the surface intermittently and is not captured by any simple formula; and the film-boiling correlation uses vapour properties at a single temperature, which is a simplification over a range of a thousand degrees. Pool boiling on a horizontal plate is also only one geometry. Water flowing through a heated tube, as in a boiler or reactor, has a critical heat flux that depends on the flow rate, the pressure and how much of the water has already turned to steam, and designers use large tabulated databases rather than a single formula for it.

The pressure matters too. The figures are for one atmosphere. At higher pressure the vapour is denser and carries more heat per unit volume, and the critical flux rises, peaking at about a third of water’s critical pressure before falling to zero at the critical point, where liquid and vapour become indistinguishable and the two become one. Pressurised-water reactors run at a hundred and fifty atmospheres partly for this reason.

What the pictures cannot show #

The figures show time-averaged fluxes on a surface of uniform temperature, and cannot show the violence of the processes averaged over. Nucleate boiling is a rapid succession of bubbles forming, growing to a millimetre or so in milliseconds and leaving, with the surface temperature under each bubble dipping as the thin layer of liquid beneath it evaporates. Transition boiling is chaotic, with liquid touching and leaving each patch of surface many times a second. Film boiling is a smooth film rippling with Rayleigh–Taylor waves that release bubbles at regular spacings. And the surface’s own properties — its roughness at the micrometre scale, its wettability, its thermal conductivity — change the curve in ways no correlation for the fluid alone can include.

Still open: how to raise the crisis #

Because the critical heat flux limits how much heat a surface can shed, raising it is worth a great deal: more power from a reactor, smaller cooling systems for processors and power electronics. Surfaces engineered at the micrometre and nanometre scale — porous coatings, arrays of tiny posts, patterns of wetting and non-wetting patches — have raised the crisis flux of water by factors of two or more in laboratory tests, apparently by wicking liquid back to dry spots and by controlling where bubbles nucleate. Why particular structures work as well as they do, whether the gains survive long operation and fouling, and whether a surface can be designed to delay the crisis by a predictable amount, are questions of active research in which the physics of the traffic between vapour and liquid at a surface is still being worked out.

The habit worth carrying away is to ask whether the thing that carries a flux can also block it. Vapour carries heat away from a boiling surface better than anything else, until there is so much of it that it separates the surface from the water — so the heat flux peaks at about 1.1 MW/m² twenty degrees above boiling and then falls fifty-fold as the surface gets hotter, and a surface held at set power past the peak jumps a thousand degrees. The drop that dances on the hottest pan is standing on the same film that melts a boiler tube.